Virtual Element Methods Without Extrinsic Stabilization

.Virtual element methods (VEMs) without extrinsic stabilization in an arbitrary degree of polynomial are developed for second order elliptic problems, including a nonconforming VEM and a conforming VEM in arbitrary dimension. The key is to construct local \(H(\operatorname{div})\)-conforming macro finite element spaces such that the associated \(L^2\) projection of the gradient of virtual element functions is computable, and the \(L^2\) projector has a uniform lower bound on the gradient of virtual element function spaces in the \(L^2\) norm. Optimal error estimates are derived for these VEMs. Numerical experiments are provided to test the VEMs without extrinsic stabilization.Keywordsvirtual elementstabilizationmacro finite elementnorm equivalenceerror analysisMSC codes65N1265N2265N30